Congruency: Congruent Triangles

128. AAS Congruence Criterion · Two angles and a side lock the triangle

Two angles and a non-included side determine a triangle uniquely (AAS).

ABPQR∠B∠C∠Q∠RACPR≅≅∠A = 45°∠A = 45°∠B = 90°∠B = 90°∠P = 45°∠P = 45°∠Q = 90°∠Q = 90°C
AAS stands for Angle-Angle-Side. Two triangles are congruent if two angles AND one side that is NOT between those angles of one triangle equal the corresponding parts of the other. The key idea: once any two angles of a triangle are fixed, the third is forced — because the three angles always sum to 180°. So knowing two angles tells you all three. Add one matching side (anywhere — included or not) and the triangle's size and shape are completely determined. AAS effectively reduces to ASA: compute the third angle, and you have two angles with their included side. Symbolically: if ∠B = ∠Q, ∠C = ∠R, and AC = PR, then ΔABC ≅ ΔPQR by AAS. Contrast with AAA — same angles alone only fix the *shape*, not the *size*; you can scale a triangle up or down without changing any angle. AAS works precisely because the one matching side pins the size.

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Selina ICSE: Congruency: Congruent Triangles

What this lesson covers

Try to break it

AAS holds because triangles have a built-in constraint — the angle-sum rule. Once two angles are fixed, the third is automatic; that's why "Angle-Angle-Side" is really "all-three-angles-plus-a-side" in disguise. Strip away the side and you are left with only AAA — which fixes shape but not size (a small triangle and a giant one can have identical angles). The matching side pins the scale and makes AAS a valid congruence criterion.

How you build it

Construct a triangle given AAS data.

  • Draw segment AC of the given length.
  • At vertex A, construct an angle equal to 180° - (∠B + ∠C).
  • At vertex C, construct an angle equal to ∠C.
  • Mark the intersection of the two rays as vertex B.

The proof, step by step

Prove that two angles and a non-included side make the triangles congruent (AAS).

  • Given: ∠B = ∠Q, ∠C = ∠R, and AC = PR (non-included side).
  • By Angle Sum Property: ∠A + ∠B + ∠C = 180° and ∠P + ∠Q + ∠R = 180°.
  • Substitution: ∠A = 180° - (∠B + ∠C) = 180° - (∠Q + ∠R) = ∠P. So, ∠A = ∠P.
  • Now we have ∠A = ∠P, AC = PR, and ∠C = ∠R. This matches ASA conditions.
  • Therefore, ΔABC ≅ ΔPQR by ASA Congruence Rule.

Worked example

In ΔABC and ΔPQR, ∠B = ∠Q = 50°, ∠C = ∠R = 80°, and AC = PR = 6 cm. Which congruence test proves ΔABC ≅ ΔPQR?

Two angles (∠B, ∠C) and a non-included side (AC) are equal to the corresponding parts of ΔPQR. This satisfies the AAS congruence condition. Hence, ΔABC ≅ ΔPQR by AAS.

  • A) AAS — correct
  • B) SAS
  • C) ASA
  • D) SSS
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